Drag mesh by single vertex/point
In order to rotate around desired vertex / vertices you can make choose them as origins of transform and change Pivot Point.

Rotate around active


Enter Edit mode of the object.
Select everything that should be rotated. To select entire mesh select one of its vertices and press Ctrl + L to select linked  edges / faces.
Deselect desired vertex which should become origin point for this moment, and select it again. This will make it active.
In 3D View header change Pivot point to Active Element or press Alt + . (period). Rotate mesh.




Rotate around 3D Cursor

You can set 3D Cursor to that vertex and rotate around it. 
This method might be more flexible because you can use it in Object mode.


Enter Edit mode.
Select what you want to be origin point. Press Shift+S > Cursor to Selected.
Change Pivot point to 3D Cursor in 3D View header (or press . (period)).
Select what should be rotated, and rotate (if several objects should be rotated go to Object mode in this step).




See also related question:


Rotate object on axis between two points